标题： 高精度线性最小化不比投影慢 显示英文标题

标题： High-precision linear minimization is no slower than projection

摘要： 此注释证明，对于所有紧致凸集，可以通过一次投影评估和一个标量-向量乘法进行高精度线性最小化。 因此，如果 $\varepsilon$-近似线性最小化至少需要$L(\varepsilon)$个实向量算术操作，且投影需要$P$个操作，则$\mathcal{O}(P)\geq \mathcal{O}(L(\varepsilon))$是有保证的。 该概念通过示例、显式误差界以及多面体集的精确线性最小化结果进行了阐述。 显示英文摘要

摘要： This note demonstrates that, for all compact convex sets, high-precision linear minimization can be performed via a single evaluation of the projection and a scalar-vector multiplication. In consequence, if $\varepsilon$-approximate linear minimization takes at least $L(\varepsilon)$ real vector-arithmetic operations and projection requires $P$ operations, then $\mathcal{O}(P)\geq \mathcal{O}(L(\varepsilon))$ is guaranteed. This concept is expounded with examples, an explicit error bound, and an exact linear minimization result for polyhedral sets.